First of all, if you’re taking the November test, good luck. If you’re prepared well, you have nothing to fear. Get in there and beast it.
In the figure above, B is the midpoint of AC and the center of the green circle. All other labeled points are also the centers of circles. If the green area is 9π, what is the area of the circle with center E?
I actually cackled sinisterly when I made this diagram. Good luck! 🙂
UPDATE: Congrats to “emolano82” for getting it first. Solution below!
There’s no “shortcut” here, other than to rub your hands together and say “OH YEAH!” like Macho Man Randy Savage. If you’re not afraid of a bunch of circles, you’ll be OK.
The key here is to take it one circle at a time. Start with the green one. If its area is 9π, then it has to have a radius of 3, right? And a diameter of 6. Which is important because…
It’s the radius of the red circle. So…so far we know BC = 3, and that the radius of the red circle above is 6. So EC = 6, because E is on the circle, and C is its center.
Looky looky! That’s a right triangle. The short leg is 3, and the hypotenuse is 6. So you can pythagorize, or you can remember your special rights and know that this is a 30°-60°-90° right off the bat, so the long leg must be 3√3. EB = 3√3.
You can do the same thing below; BD = 3√3. So ED, which is the radius of the circle with center E, must be 6√3.
To find the area: